Omitted Variable Bias and Suppressor Variables
You have two OLS regressions on the same dataset:
- Model 1: $Y = b_1 X_1 + \epsilon$ (simple regression)
- Model 2: $Y = b_2 X_1 + \beta_2 X_2 + \epsilon$ (multiple regression)
You are told that $\text{corr}(X_1, X_2) > 0$ and $\text{corr}(Y, X_2) = 0$.
Which coefficient is larger, $b_1$ or $b_2$? Prove it.
Hints
- Think about what happens to the coefficient on $X_1$ when you omit a correlated regressor -- there is a classical formula for this bias.
- Apply the omitted variable bias formula: $b_1 = b_2 + \hat{\gamma} \cdot \beta_2$, where $\hat{\gamma}$ is from regressing $X_2$ on $X_1$. Determine the sign of each factor separately.
- Use the OLS normal equations to relate $\text{Cov}(Y, X_2) = 0$ to the sign of $\beta_2$ in Model 2. You will find $\beta_2 = -b_2 \cdot \text{Cov}(X_1, X_2) / \text{Var}(X_2) < 0$, which pins down the direction.
Worked Solution
How to Think About It: The key tool here is the omitted variable bias (OVB) formula, which tells you exactly how a coefficient changes when you add or remove a regressor. Think of it this way: Model 1 omits $X_2$. If $X_2$ is correlated with $X_1$, then in Model 1, $X_1$ is "doing double duty" -- it is picking up its own effect on $Y$ plus whatever effect $X_2$ has on $Y$ that leaks through their correlation. The question is: does omitting $X_2$ make $b_1$ bigger or smaller than $b_2$?
The answer turns on the sign of $\beta_2$ in Model 2. At first glance, $\text{corr}(Y, X_2) = 0$ looks like it means $X_2$ has no effect on $Y$. But that is the *marginal* correlation, not the *partial* effect. Once you condition on $X_1$, the partial effect of $X_2$ on $Y$ can be non-zero -- and in this case, it must be negative. That is what makes $X_2$ a suppressor variable.
Key Setup:
The omitted variable bias formula relates the two coefficients:
$$b_1 = b_2 + \hat{\gamma} \cdot \beta_2$$
where $\hat{\gamma}$ is the coefficient from the auxiliary regression of $X_2$ on $X_1$ (i.e., how much $X_2$ changes per unit of $X_1$).
This formula is exact in OLS. It says: the coefficient on $X_1$ in the short regression equals the coefficient in the long regression, plus a bias term equal to (how $X_2$ loads on $X_1$) times (the effect of $X_2$ on $Y$ in the long model).
Step 1 -- Sign of $\hat{\gamma}$:
Since $\text{corr}(X_1, X_2) > 0$, we have $\hat{\gamma} > 0$.
Step 2 -- Sign of $\beta_2$:
This is the subtle step. The unconditional correlation $\text{corr}(Y, X_2) = 0$ means:
$$\text{Cov}(Y, X_2) = 0$$
In Model 2, the fitted $Y$ is $b_2 X_1 + \beta_2 X_2$. The OLS normal equations give:
$$\text{Cov}(Y, X_2) = b_2 \text{Cov}(X_1, X_2) + \beta_2 \text{Var}(X_2) = 0$$
Solving for $\beta_2$:
$$\beta_2 = -b_2 \cdot \frac{\text{Cov}(X_1, X_2)}{\text{Var}(X_2)}$$
Since $\text{Cov}(X_1, X_2) > 0$ and $\text{Var}(X_2) > 0$, we get $\beta_2 < 0$ (assuming $b_2 > 0$, which is the natural assumption; if $b_2 = 0$ then $b_1 = b_2$ trivially).
Step 3 -- Direction of the bias:
Substituting back into the OVB formula:
$$b_1 = b_2 + \underbrace{\hat{\gamma}}_{> 0} \cdot \underbrace{\beta_2}_{< 0} = b_2 - (\text{positive quantity})$$
Therefore $b_1 < b_2$.
Answer: $b_1 < b_2$. The simple regression coefficient is smaller than the multiple regression coefficient. Omitting $X_2$ introduces a downward bias on $b_1$, because $X_2$ acts as a suppressor -- it is positively correlated with $X_1$ but has a negative partial effect on $Y$ (to offset the positive contribution of $X_1$ flowing through their correlation).
Intuition
This problem illustrates a concept that trips up a lot of quants: zero marginal correlation does not mean zero partial effect. The variable $X_2$ has no raw correlation with $Y$, but once you control for $X_1$, it has a negative effect. This is the classic suppressor variable setup. In practice, suppressors show up in macro factor models all the time -- a factor can appear useless in a univariate regression but turn highly significant (and with the opposite sign you might expect) once correlated factors are included. The takeaway: always look at partial effects in multivariate regressions, not just raw correlations.
The omitted variable bias formula $b_1 = b_2 + \hat{\gamma} \cdot \beta_2$ is one of the most useful tools in applied econometrics. It tells you exactly how to decompose the difference between coefficients across nested models. In strategy research, this comes up when you evaluate whether adding a control variable to a predictive regression inflates or deflates your signal coefficient -- and by how much. A common mistake is assuming that if $\text{corr}(Y, X_2) \approx 0$, adding $X_2$ to the model barely changes anything. This problem shows that is wrong: when $X_2$ is correlated with $X_1$, its inclusion can substantially shift the $X_1$ coefficient even if $X_2$ appears uninformative on its own.